ORAU Team 

Dose Reconstruction Project for NIOSH

Technical Information Bulletin – Bayesian Methods for 

Estimation of Unmonitored Y-12 External Penetrating 

Doses with a Time-Dependent Lognormal Model 

Document Number: 

ORAUT-OTIB-0015 

Effective Date: 09/09/2004 

Revision No.: 00 

Controlled Copy No.: ______ 

Page 1 of 11

Subject Expert: Peter G. Groer 

Approval: Signature on File Date: 09/09/2004

 Judson L. Kenoyer, Task 3 Manager

Concurrence: Signature on File Date: 09/09/2004

 Richard E. Toohey, Project Director

Approval: Signature on File Date: 09/09/2004

 James W. Neton, Associate Director for Science 

Supersedes: 

None
Effective Date: 09/09/2004 Revision No. 00 Document No. ORAUT-OTIB-0015 Page 2 of 11 

RECORD OF ISSUE/REVISIONS 

ISSUE 

AUTHORIZATION 

DATE 

EFFECTIVE 

DATE 

REV. 

NO. DESCRIPTION 

Draft 08/24/2004 00-A New technical information bulletin for the 

Bayesian Methods for Estimation of Unmonitored 

Y-12 External Penetrating Doses with a TimeDependent Lognormal Model. Initiated by Peter 

G. Groer. 

09/09/2004 09/09/2004 00 First approved issue. Initiated by Peter G. Groer. 
Effective Date: 09/09/2004 Revision No. 00 Document No. ORAUT-OTIB-0015 Page 3 of 11 

ACRONYMS AND ABBREVIATIONS 

LOD limit of detection 

MLE maximum likelihood estimate 

mrem millirem 
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1.0 INTRODUCTION

This document describes Bayesian procedures in general, discusses a Bayesian regression and 

prediction analysis of external penetrating dose data for Y-12 with a time-dependent lognormal model, 

and an estimated a scale factor for imputation of unmonitored doses for an individual. The original 

data set (Watkins et al. 2004) for the regression and prediction analysis incorporated data from 1950 

to 1970. This analysis only used the data from 1956 to 1961. For the regression analysis, all doses 

less than 15 mrem were replaced by 15 mrem. For the estimation of the scale factor, all observations 

less than or equal to 30 mrem [i.e., the reported limit of detection (LOD)] were treated as left-censored 

observations. This required some modification of the likelihood construction. 

Bayesian methods work for all sizes of data sets, large or small. For the large data set (Watkins et al. 

2004) considered here, maximum likelihood estimates (MLEs; Frome and Watkins 2004) are 

appropriate and produce numerically practically identical results. This is demonstrated by a 

comparison of Bayesian estimates and the MLEs for the regression analysis and the scaling 

procedure. 

All analyses assumed that the doses in the original data set (Watkins et al. 2004) were precise with 

the exception of the doses that were treated as left censored. 

2.0 REGRESSION AND PREDICTION ANALYSIS

2.1 APPROACH 

The analysts estimated the parameters log (β) and μ for the following time-dependent lognormal 

model using Bayesian methods: 

( )

( ) () () 2

2

log log

2 1 |,, 2

y t

fy e

y

β μ

ο β μσ

πσ

− + −

=

The probability density shown above was used to construct the likelihood p(D | log (β), μ, σ) (D means 

data used). According to Bayes’ theorem the joint posterior density p(log (β), μ, σ | D) of the model 

parameters is proportional to the likelihood if uniform priors are used: 

p(log (β), μ, σ | D) ∝ p(D | log (β), μ, σ) 

Because the parameters of greatest interest are log (β) and μ, the joint density p(log (β), μ, σ | D) was 

first integrated over σ. This integration can be done analytically if no left-censored observations are 

present, and it gives the following expression for the joint posterior density p(log (β), μ | D) for log (β) 

and μ: 

( ) ( ) ( ) ()

40

1

1 /2 40

1 1

log , | log log

n k n

l l

ik k

i k

pD y t βμ β μ =

 

− −      

= =

∑  

= −+     ∑∑

Where k labels the forty quarters from 1956 to 1965, and lk is the number of observations yik in each 

quarter. The time tk is measured in quarter years starting from 1956. 
Effective Date: 09/09/2004 Revision No. 00 Document No. ORAUT-OTIB-0015 Page 5 of 11 

Prediction of unmonitored doses is accomplished with the predictive density. This is a strictly 

Bayesian concept that permits quantitative consideration of parameter uncertainty when making 

predictions. For the model under consideration the predictive density p(yf | D) for a unmonitored 

observation yf is given by: 

( ) | |,, ,, | ( ) ( ) f f p y D f y p Dd

θ

= β μσ β μσ θ ∫

The integration is over all model parameters θ = {β, μ, σ}. Section 2.2 shows sample plots of p(yf | D) 

for three different years. Point estimates based on maximum likelihood are compared with posterior 

expectations obtained by the Bayesian approach. 

2.2 RESULTS 

Figure 1 shows the joint posterior density p(log (β), μ | D) after integration over σ, and Figures 2 and 3 

show the marginal posterior densities of the model parameters log (β) and μ, respectively. 

Figure 1. Joint posterior density p(log (β), μ | D) (4.1 < log (β) < 4.35, 0.07 < μ < 0.12). 

The posterior expectations (E) and standard deviations (SD) of the parameters are: 

E(log (β)) = 4.2159, SD(log (β)) = 0.02671 

E(μ) = 0.09281, SD(μ) = 0.00453 

This agrees with the estimates obtained with maximum likelihood methods in Frome and Watkins 

(2004), which are: 

E(log (β)) = 4.2159, SD(log (β)) = 0.02668 

E(μ) = 0.09281, SD(μ) = 0.00452 
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Figure 2. Posterior density p(log (β) | D). 

Figure 3. Posterior density p(μ | D). 

A detailed comparison for 1956 (i.e., t = 0) gave the following results: 

E(log(yf) | D) = 4.2174 (which agrees with E(log (β)) from above because t = 0.) 

SD(log(yf) | D) = 0.9867 

E(yf | D) = 110.32 

SD(yf | D) = 142.37 

These results imply a geometric mean of GM(yf | D) = 67.57 and a geometric standard deviation of 

GSD(yf | D) = 2.69. 

The corresponding estimates based on maximum likelihood techniques (Frome and Watkins 2004) 

are: 

E(log(yf) | D) = 4.2159 = E(log (β)) from above because t = 0) 

SD(log(yf) | D) = 0.9916 
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E(yf | D) = 110.78 

SD(yf | D) = 143.31 

Again, these results imply a geometric mean GM(yf | D) = 67.75 [sic] and a geometric standard 

deviation GSD(yf | D) = 2.69. 

Figure 4 shows that the exact and the approximate lognormal predictive (dashed) densities are 

practically indistinguishable. 

p(yf | D) 

yf

Figure 4. Exact (—) and approximate (---) predictive density for yf.

The right tail probability p(yf > 346.2 | D) = 0.05 also agrees with the tail probability calculated with the 

approximate predictive density. 

Figures 5 and 6 show two additional plots of p(yf | D) for 1951 and 1957, respectively. Some point 

estimates for the same years are: 

Year E(log(yf)) SD(log(yf))

1951 4.6812 0.9928 

1957 4.1243 0.9920 

The corresponding MLEs are: 

Year MLE(log(yf)) SD(log(yf))

1951 4.6800 0.9914 

1957 4.1231 0.9918 
Effective Date: 09/09/2004 Revision No. 00 Document No. ORAUT-OTIB-0015 Page 8 of 11 

p(yf | D) 

yf

Figure 5. Plot of p(yf | D) for 1951. 

Figure 6. Plot of p(yf | D) for 1957. 

2.3 CONCLUSIONS 

For this large data set, without censored observations, the difference between exact predictive 

densities and their lognormal approximations is negligible (see Figure 4). For the data set with left 

censoring at the reported LOD of 30 mrem, some difference between exact and approximate densities 

would be expected, dependent on the number of censored observations. The magnitude of the 

difference must be evaluated numerically because the likelihood terms for observations censored at, 

for example, d, shown below prevent symbolic integration over the parameter σ: 
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3.0 BAYESIAN ESTIMATION OF A SCALE FACTOR FOR IMPUTING UNMONITORED 

DOSES

3.1 APPROACH 

For a Y-12 worker, whose potential for exposure before 1961 is judged to be the same as after 1961, 

it is possible to estimate a scale factor φ based on the worker’s post-1961 dose records. This 

parameter φ measures the discrepancy between the worker’s dose and the population dose after 

1961 (Frome and Groer 2004). 

The Bayesian estimation of φ, defined above, starts again with a likelihood similar to L(φ|d,μ,σ)

(Frome and Watkins 2004). The likelihood used for the Bayesian analysis is slightly different because 

dt = 0 is treated as a left-censored observation whenever it occurs. Using a subscript c (meaning 

censored) to differentiate it from L(φ|d,μ,σ) (Frome and Watkins 2004), the likelihood is now given by 

the following expression: 

( ) ( ) ( ) ( )

17 2 log

1

1 |,, i i d

C C

i i

Ld e F μ φ φ μσ φ

σ

− −−

=

 

∝ ×     ∏

Constants not depending on φ have been dropped for brevity, and i indicates all quarters for which 

di > 0. For the example presented below, Fc (φ) is a product of three terms of the generic form: 

( ) (30) | , 2

t

C tt

t

log f Erfc μ φ

φμσ

σ

  + − =    

where μt and σt are replaced by the corresponding numerical estimates for the quarters for which 

dt = 0. The following integral defines the right-hand side of fc above: 

( ) ( ) ( ) 2

2

log 30

2

0

1 30

2

t t

t

d

t

t

t t t

log e dd Erfc d

μ φ

σ μ φ

σ σ

− − −   + − =     ∫

The upper limit of the integral (30 mrem) is the reported LOD mentioned above. 

The posterior density of φ, p(φ|D), is proportional to Lc(φ|d,μ,σ) given above (D means the dose data 

of the individual in the example considered here). 

3.2 RESULTS 

Figure 7 shows a graph of the posterior density of φ. The expectation [E(φ|D)] is 0.492, and the 

variance [Var(φ|D)] is 0.0369. 

A Bayesian analysis treating dt = 0 as left-censored observations was applied to check the accuracy 

of the approximation of replacing non-detects by the conditional expectation of y, yo

t (Frome and 

Groer 2004). 

The slight difference between E(φ|D) and the MLE result (Frome and Watkins 2004) is caused by the 

two different methods used to deal with non-detects. 
Effective Date: 09/09/2004 Revision No. 00 Document No. ORAUT-OTIB-0015 Page 10 of 11 

φ

Figure 7. Posterior density p(φ|D). 

p(φ|D) 
Effective Date: 09/09/2004 Revision No. 00 Document No. ORAUT-OTIB-0015 Page 11 of 11 

REFERENCES 

Frome, E. L., and P. G. Groer, 2004, Individual Dose Adjustment Procedure for Y-12 Dose 

Reconstruction, ORAUT-TIB-0013, Revision No. 00-A, Oak Ridge Associate Universities, Oak 

Ridge, Tennessee. 

Frome, E. L., and J. P. Watkins, 2004, Statistical Analysis of Data with Non-detectable Values, 

ORNL/TM-2004/146 (under review), Oak Ridge National Laboratory, Oak Ridge, Tennessee. 

Watkins, J. P., G. D. Kerr, E. L. Frome, W. G. Tankersley, and C. M. West, 2004, Historical Evaluation 

of the Film Badge Dosimetry Program at the Y-12 Facility in Oak Ridge, Tennessee, (Part 1 – 

Gamma Radiation), ORAU Technical Report 2004-0888, Oak Ridge Associated Universities, 

Oak Ridge, Tennessee.